Once again it is time to roll over the previous discussion thread, which has become rather full with comments. The paper is nearly finished (see also the working copy at this subdirectory, as well as the rest of the directory), but several people are carefully proofreading various sections of the paper. Once all the people doing so have signed off on it, I think we will be ready to submit (there appears to be no objection to the plan to submit to Algebra and Number Theory).
Another thing to discuss is an invitation to Polymath8 to write a feature article (up to 8000 words or 15 pages) for the Newsletter of the European Mathematical Society on our experiences with this project. It is perhaps premature to actually start writing this article before the main research paper is finalised, but we can at least plan how to write such an article. One suggestion, proposed by Emmanuel, is to have individual participants each contribute a brief account of their interaction with the project, which we would compile together with some additional text summarising the project as a whole (and maybe some speculation for any lessons we can apply here for future polymath projects). Certainly I plan to have a separate blog post collecting feedback on this project once the main writing is done.
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2 November, 2013 at 5:37 pm
pigh3
A few more typos:
is now 
according to the new notation in Def 2.14
is probably missing “1+”.
. Probably better to use 
to be consistent with later and with other sections.
1. Table 1 (p8),
2. p74, immediately after proof of Lemma 5.3, “We will can prove”: “can” can probably be changed to “now”.
3. Bottom of p74, definition of
4. p81, 4th display & 2 lines above it, there are 2 uses of
5. p83, 1st paragraph of sec 6.3,”We with a” is missing a verb like “begin”.
6. p 170, Ref 22, missing \”a in Birkhauser.
7. p 172, Ref 75, Yildirim may be changed to Y\i ld\i r\i m.
3 November, 2013 at 3:21 am
Eytan Paldi
In the definition of
(theorem 2.16), a space is needed between the subscript 
and 
.
3 November, 2013 at 12:23 pm
Dan Goldston
I think that footnote 5 on page 13 isn’t correct. Bombieri and Davenport’s proof does not give (1.1) on EH, in fact assuming EH does not help in their proof at all. The problem is that the major arc approximation used there becomes worse for Farey arcs with denominator larger than N^(1/2) because the arcs become so short. Even if one replaces Bombieri and Davenport’s arc widths 1/Q^2 with 1/qQ this does not help. One can generalize their proof as I did in http://www.math.sjsu.edu/~goldston/article17.pdf and obtain (1.1) on EH for both Lambda and the sieve weight Lambda_Q (stated on the second page of that paper,) but I’ve never believed that assuming EH for an object like Lambda_Q is worth anything. I tried to get (1.1) from EH for Lambda and for Mu, but never succeeded.
3 November, 2013 at 10:49 pm
Emmanuel Kowalski
Thanks for the correction! Actually what you write is pretty much what I had read in your paper when writing my Bourbaki report on your work with Pintz and Yildirim (http://www.math.ethz.ch/~kowalski/goldston-pintz-yildirim.pdf, see Remarque 3.1, which mentions your paper and that “a certain form of EH would imply…”), but when I wrote the footnote in the Polymath8 paper I just vaguely remembered the conclusion (and had forgotten the rest…)
I think it’s simpler here to delete the footnote (in the earlier account, it was useful as a way of showing a faint historical trail, and emphasizing the link between the original motivation of Bombieri for the B-V theorem and gaps between primes.)
4 November, 2013 at 12:43 pm
Emmanuel Kowalski
I am reviewing typei-ii.tex (again, a bit slower than I’d hoped).
Here’s a question concerning the reduction to exponential sums (with references to the current newgap.pdf in the main folder): it seems to me that when completing the sum, we are doing as if (in (7.22) and later) the modulus of the sum that we complete is q_1q_2r instead of [q_1,q_2]r (see the value of H in (7.23)). Of course this is irrrelevant in terms of the length of the dual sum over h, which can be increased, but it affects the factor 1/H=M/q (roughly) from Lemma 6.9 (ii), which seems to be bigger than what we write by a factor q_0.
If I understood this right, this means we need a target (7.27) smaller by a factor 1/q_0, i.e., a RHS x^{-\eps}Q^2NR/q_0^2.
From a quick read-through of the remainder ot Section 7 and of Section 10, this is actually something we prove, because the exponents of q_0 in all conditions to check are always strong enough (and of course it doesn’t affect the generic case q_0=1). But before making the changes to (7.27), I’d like to make sure that I am not missing something obvious here…
4 November, 2013 at 1:17 pm
Terence Tao
Oops, you are right, this extra factor of
is indeed missing, but we have enough powers of 
to spare to deal with this. (I checked Andrew Granville’s version of the argument and it looks to be OK because he does the completion of sums a little differently.) I’ll make sure to look out for the powers of q_0 when you’re done with this section (and with Section 10).
5 November, 2013 at 8:04 am
Emmanuel Kowalski
I am getting confused now with the next step (the Type II estimate) — on the last line of Page 109 of the main folder version, the powers of q_0 are tight (q_0^{-2} on both sides), so if we need a better saving in q_0, it doesn’t work outright (precisely, we need q_0^{-4} after squaring, but we only gain one q_0 from the smaller size of H).
This doesn’t seem to be problematic for Type I estimates on the other hand.
The problem comes from the absence of gain in q_0 in the second term of Proposition 7.8, which however seems wasteful. (I haven’t checked yet in Zhang’s treatment or Andrew’s to compare…)
5 November, 2013 at 9:49 am
Terence Tao
Oh dear, this is a non-trivial issue. For non-zero
, I think it can be repaired by exploiting more fully the constraint 
in the phase (7.24), which after fixing 
constrains 
to a single residue class modulo 
. This gives an additional factor of 
saving in (7.28) which seems to give enough room to fit in the loss of 
you pointed out.
The
case remains a problem. Of course, by restricting to this case, one saves a factor of 
, but the way things are set up right now, 
is tiny (as small as 
) and so this does not directly help us. Zhang sets 
to be larger than this in the Type II case (something more like 
, if I remember correctly), which would solve this problem but worsen the Type II numerology a little bit, which would be annoying.
The other thing we could do is try to eliminate the
case earlier, before completion of sums. This is actually what Zhang does, but the price one pays for this is that the residue class 
that one initially works with can no longer be completely arbitrary, but has to obey a “controlled multiplicity” condition, basically for any 
, one cannot have more than 
moduli 
for which 
.
I think the issue may also affect Andrew’s version of the argument; he correctly tracked the powers of
(which he calls 
) all the way through to the third display of page 49, which I have not yet checked carefully. (Also there is a tiny typo in that paper: in the penultimate display of page 47, 
should be 
rather than 
.) I have to run for now, but will return to this issue later.
5 November, 2013 at 10:21 am
Emmanuel Kowalski
I think Andrew mentions something about (the analogue of) Prop. 7.8 not being optimal in terms of q_0 (or is it about another lemma? I have to check).
I will also continue thinking about this…
5 November, 2013 at 11:35 am
Terence Tao
OK, I think I have a fix which is not too bad. It involves treating the
case separately, and using the additional constraint on 
in the 
case. The precise changes are as follows
1. After the paragraph containing (7.20), declare that we will treat the
contribution first. Here we do not split off an X term, and simply aim to bound
in magnitude by
.
We write
and the divisor bound to crudely bound this quantity by
By the Chinese remainder theorem, the l summation is![O( \frac{x}{r[q_1,q_2]} )](https://s0.wp.com/latex.php?latex=O%28+%5Cfrac%7Bx%7D%7Br%5Bq_1%2Cq_2%5D%7D+%29&bg=ffffff&fg=545454&s=0&c=20201002)
(one has to note that 
, which can be deduced from (7.2), (7.12), (7.13) with plenty of room to spare), so we now have
which sums to
, which is acceptable since 
.
For the rest of section 7 we insert the condition
.
2. Now that k is nonzero, we can insert an additional factor of
on the RHS of (7.27), thanks to Lemma 1.6. We’ll need this factor later.
3. In Section 7.3, we save the constraint
from (7.24) and combine it with the 
weight. Observe that as 
is coprime to 
, this constraint restricts 
to at most 
residue classes modulo 
. If we use (1.5) instead of (1.4), we can now insert a factor of 
in (7.28) and the preceding display after inserting the 
weight (and removing the equality symbol in the display before (7.28)). This gains us an extra factor of 
in the rest of the argument which should compensate for the loss of 
that you noted previously.
4. Some minor modifications may need to be made to the Type I arguments (either we save the factor of
as in the Type II estimates, or we simply throw this improvement away and take the loss of 
).
5 November, 2013 at 11:39 am
Emmanuel Kowalski
I’ll check and incorporate this version of the argument tomorrow. At first sight it certainly seems reasonable (and philosophically it would be strange if the numerology with gcd q_0 >1 does not conform to that with q_0=1…)
5 November, 2013 at 1:44 pm
Gergely Harcos
Terry’s fix looks good to me. For the new version of (7.28) with the extra factor
, we need that 
, but this follows (7.12), (7.13), and the conditions in Theorem 7.7.
6 November, 2013 at 9:09 am
Emmanuel Kowalski
I’ve put up in the subfolder what I have done so far with typei-ii.tex, which is up to the end of Section 7.4. I will begin the Type I estimates tomorrow.
6 November, 2013 at 11:40 am
Terence Tao
I looked through it and it seems fine to me. Some minor comments:
page 97, after the discussion of the off-diagonal contribution: “there can be cancellation between these non-negative terms” should probably be “there cannot be any cancellation between these non-negative terms”. Also, in footnote 18, I was not quite clear as to what “repeating the computation” meant… I guess you are trying to say that we should not be too fixated on the operator norm per se, but rather on the computational techniques used to estimate such norms?
around (7.4): ironically, with our new fix, the case k=0 that we considered “for simplicity” becomes the case for which we do NOT apply the method indicated! But perhaps this is OK as long as we admit that we are “lying” when giving this oversimplified presentation.
In (7.29), a factor of
is missing on the RHS. The definition of H here is now very slightly different from that in the previous section, but they are equivalent up to constants; actually, for consistency, we may wish to take H equal to 
before Remark 7.7, rather than ![x^\varepsilon [q_1,q_2] r M^{-1}](https://s0.wp.com/latex.php?latex=x%5E%5Cvarepsilon+%5Bq_1%2Cq_2%5D+r+M%5E%7B-1%7D&bg=ffffff&fg=545454&s=0&c=20201002)
.
6 November, 2013 at 9:41 pm
Emmanuel Kowalski
In footnote 18, I wanted to mean that, in some sense, we repeat three or four (or more…) times the same technique, more or less explicitly, instead of having stated an abstract lemma and applying it. (It is not a very important remark so it might be removed.)
I’ll take care of the other corrections. Actually, it seems the exponential sum estimate is now even better than strictly needed in terms of q_0 when keeping track of the support in n for k non-zero (restricting to few congruence classes modulo q_0) but I haven’t propagated this improvement to the main statement of Theorem 7.8 (of the ek version).
6 November, 2013 at 10:52 pm
Emmanuel Kowalski
Actually, what I wrote is absurd, I forgot the squaring-step so the gain we need comes from the n sum in both factors of Cauchy-Schwarz, but I forgot to write it on the second side.
7 November, 2013 at 9:43 am
Emmanuel Kowalski
The first Type I estimate is now done in the ek folder. I will now work on the last…
7 November, 2013 at 12:22 pm
Anonymous
Bibliography:
* [6] and [16]: It should be an n-dash instead of a hyphen when indicating a (page) range
* [25]–[29]: “Ph.Michel” –> “Ph. Michel”
* [43] and [52]: Remove “pp.”
* [69]: A comma before the page range
* [78]: “G.N.” –> “G. N.”
11 November, 2013 at 1:45 am
Emmanuel Kowalski
Corrections done in the ek folder.
11 November, 2013 at 4:47 am
Anonymous
Missing: “no. 3-4” –> “no. 3–4” in [6].
8 November, 2013 at 4:36 am
Emmanuel Kowalski
I am done reviewing Section 7. The only other remark I had is that I think the middle-condition (which is always the same) in the type I cases seems to involve sigma because one needs a lower-bound for N, namely it should be
8\varpi + 3\delta< 1/2-\sigma
instead of
20\varpi+6\delta<1.
I might have misunderstood, but in any case this condition is always weaker than the first condition, so it doesn't change anything.
The gain of q_0 was also needed for the second Type I estimate, but not (as far as I could see) for the first.
I now hope to read Section 8 fairly quickly…
9 November, 2013 at 6:03 pm
pigh3
A few more, typos and style suggestions:
-> 
?
-> 
“Lemma 6.9 and Proposition 6.10”.
1. p 77, line -7 of Proof of Lemma 5.4, Siegel-Walfisz theorem -> Siegel-Walfisz property.
2. p 78, line 9, Type III needs \sigma in bracket.
3. p 78, line 10, direclty -> directly
4. p 79, 4th line under (5.19),
5. p 79, 4 lines under previous,
[Unfortunately, I could not fully repair your item 5; it appears that < and > signs have been interpreted as HTML. You will need to use < and > instead. -T.]
[These changes have been made to the main folder files -T.]
10 November, 2013 at 4:19 pm
pigh3
[Thanks for editing, Terry. The <‘s and >’s made quite a mess. The original was quite a bit longer. I cannot fix 5, so I will try to avoid <‘s:]
5. p 79, 4 lines under previous, another , should be *?
should be 
.
6. p 84, statement and proof of Lemma 6.5, (1) and (2) can be changed to (i) and (ii) for consistency.
7. p 84, line after (6.6), b should be (b,q)?
8. p 85, line after 1st display of proof of Prop 6.6, Z/qZ should be Z/pZ?
9. p 86, 3rd line of proof of Lemma 6.8,
10. pp 89-90, statement of Prop 6.10, (i) and (ii) need to be in roman font to be consistent with other places. (Same for several places in Sec. 7)
11. p 90, line -3 of Remark 6.11, “Lemmas 6.9 and 6.10” should be “Lemma 6.9 and Proposition 6.10”.
[These changes have been made to the main folder files -T.]
10 November, 2013 at 10:54 pm
Emmanuel Kowalski
I incorporated these also in the ek subfolder.
10 November, 2013 at 12:12 pm
Emmanuel Kowalski
I have now put the deligne.tex section in the ek subfolder. The changes there are few:
(1) I added references to some more surveys (one a slightly older one of mine, and an upcoming one by Fouvry, Philippe and myself (that will appear in the Colloquio di Giorgi series, and which will be ready soon)
(2) some exponents for conductors were a bit off, I think (this is immaterial for the applications of course)
(3) I clarified a bit subsection 8.3 which might have been a bit unclear; one or two references might be useful there, which I will take care of.
(4 I wrote the section on incomplete sums as in exponential.tex, independently of the standard asymptotic convention
(5) I gave some details in Corollary 8.25 on how to sum with the coprimality condition, since this type of restriction is not directly handled by Section 6.
Onward to typeiii.tex !
17 November, 2013 at 10:58 am
Polymath8: Writing the first paper, V, and a look ahead | What's new
[…] time to (somewhat belatedly) roll over the previous thread on writing the first paper from the Polymath8 project, as this thread is overflowing with comments. […]
4 December, 2013 at 3:06 am
Anonymous
8\varpi + 3\delta< 1/2-\sigma
4 December, 2013 at 3:08 am
Anonymous
\mu_
4 December, 2013 at 3:53 am
Anonymous
What are you talking about? If you have found an error, tell where it is. No one has any chance of knowing what you are talking about.